Kelvin's circulation theorem

Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states In a barotropic ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time.[1][2] Stated mathematically: {style d'affichage {frac {mathrm {ré} Gamma }{mathrm {ré} t}}=0} où {style d'affichage Gamma } is the circulation around a material contour {displaystyle C(t)} . Stated more simply this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal.

This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations.

Contenu 1 Mathematical proof 2 Poincaré–Bjerknes circulation theorem 3 Voir également 4 Notes Mathematical proof See also: Euler equations (fluid dynamics) The circulation {style d'affichage Gamma } around a closed material contour {displaystyle C(t)} est défini par: {style d'affichage Gamma (t)=point _{C}{symbole gras {tu}}cdot mathrm {ré} {symbole gras {s}}} where u is the velocity vector, and ds is an element along the closed contour.

The governing equation for an inviscid fluid with a conservative body force is {style d'affichage {frac {mathrm {ré} {symbole gras {tu}}}{mathrm {ré} t}}=-{frac {1}{Rho }}{symbole gras {nabla }}p+{symbole gras {nabla }}Phi } where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.

The condition of barotropicity implies that the density is a function only of the pressure, c'est à dire. {displaystyle rho =rho (p)} .

Taking the convective derivative of circulation gives {style d'affichage {frac {mathrm {ré} Gamma }{mathrm {ré} t}}=point _{C}{frac {mathrm {ré} {symbole gras {tu}}}{mathrm {ré} t}}cdot mathrm {ré} {symbole gras {s}}+point _{C}{symbole gras {tu}}cdot {frac {mathrm {ré} mathrm {ré} {symbole gras {s}}}{mathrm {ré} t}}.} For the first term, we substitute from the governing equation, and then apply Stokes' theorem, Donc: {displaystyle oint _{C}{frac {mathrm {ré} {symbole gras {tu}}}{mathrm {ré} t}}cdot mathrm {ré} {symbole gras {s}}=int _{UN}{symbole gras {nabla }}temps restant(-{frac {1}{Rho }}{symbole gras {nabla }}p+{symbole gras {nabla }}Phi right)cdot {symbole gras {n}},mathrm {ré} S=int _{UN}{frac {1}{Rho ^{2}}}la gauche({symbole gras {nabla }}rho times {symbole gras {nabla }}pright)cdot {symbole gras {n}},mathrm {ré} S=0.} The final equality arises since {style d'affichage {symbole gras {nabla }}rho times {symbole gras {nabla }}p=0} owing to barotropicity. We have also made use of the fact that the curl of any gradient is necessarily 0, ou {style d'affichage {symbole gras {nabla }}fois {symbole gras {nabla }}f=0} for any function {style d'affichage f} .

For the second term, we note that evolution of the material line element is given by {style d'affichage {frac {mathrm {ré} mathrm {ré} {symbole gras {s}}}{mathrm {ré} t}}=gauche(mathrm {ré} {symbole gras {s}}cdot {symbole gras {nabla }}droit){symbole gras {tu}}.} Ainsi {displaystyle oint _{C}{symbole gras {tu}}cdot {frac {mathrm {ré} mathrm {ré} {symbole gras {s}}}{mathrm {ré} t}}=point _{C}{symbole gras {tu}}point gauche(mathrm {ré} {symbole gras {s}}cdot {symbole gras {nabla }}droit){symbole gras {tu}}={frac {1}{2}}point _{C}{symbole gras {nabla }}la gauche(|{symbole gras {tu}}|^{2}droit)cdot mathrm {ré} {symbole gras {s}}=0.} The last equality is obtained by applying gradient theorem.

Since both terms are zero, we obtain the result {style d'affichage {frac {mathrm {ré} Gamma }{mathrm {ré} t}}=0.} Poincaré–Bjerknes circulation theorem A similar principle which conserves a quantity can be obtained for the rotating frame also, known as the Poincaré–Bjerknes theorem, named after Henri Poincaré and Vilhelm Bjerknes, who derived the invariant in 1893[3][4] et 1898.[5][6] The theorem can be applied to a rotating frame which is rotating at a constant angular velocity given by the vector {style d'affichage {symbole gras {Oméga }}} , for the modified circulation {style d'affichage Gamma (t)=point _{C}({symbole gras {tu}}+{symbole gras {Oméga }}fois {symbole gras {r}})cdot mathrm {ré} {symbole gras {s}}} Ici {style d'affichage {symbole gras {r}}} is the position of the area of fluid. From Stokes' theorem, this is: {style d'affichage Gamma (t)=int _{UN}{symbole gras {nabla }}fois ({symbole gras {tu}}+{symbole gras {Oméga }}fois {symbole gras {r}})cdot {symbole gras {n}},mathrm {ré} S=int _{UN}({symbole gras {nabla }}fois {symbole gras {tu}}+2{symbole gras {Oméga }})cdot {symbole gras {n}},mathrm {ré} S} The Vorticity of a velocity field in fluid dynamics is defined by: {style d'affichage {symbole gras {oméga }}={symbole gras {nabla }}fois {symbole gras {tu}}} Alors: {style d'affichage Gamma (t)=int _{UN}({symbole gras {oméga }}+2{symbole gras {Oméga }})cdot {symbole gras {n}},mathrm {ré} S} See also Bernoulli's principle Euler equations (fluid dynamics) Helmholtz's theorems Thermomagnetic convection Notes ^ Katz, Plotkin: Low-Speed Aerodynamics ^ Kundu, P and Cohen, je: Fluid Mechanics, page 130. Presse académique 2002 ^ Poincaré, H. (1893). Théorie des tourbillons: Leçons professées pendant le deuxième semestre 1891-92 (Volume. 11). Gauthier-Villars. Article 158 ^ Truesdell, C. (2018). The kinematics of vorticity. Courrier Douvres Publications. ^ Bjerknes, V, Rubenson, R, & Lindstedt, UN. (1898). Ueber einen Hydrodynamischen Fundamentalsatz und seine Anwendung: besonders auf die Mechanik der Atmosphäre und des Weltmeeres. Kungl. Boktryckeriet. PA Norstedt & Söner. ^ Chandrasekhar, S. (2013). Hydrodynamic and hydromagnetic stability. Courier Corporation. Catégories: Equations of fluid dynamicsFluid dynamicsEquationsWilliam Thomson, 1st Baron Kelvin